Ordinary Differential Equations and Dynamical Systems Vol2 · Problem 2.5

Existence Theory

Mathematics — Calculus · jordan-form-stable-unstable-center-subspaces

Problem

Define A=[8008800088000000000000008]A = \begin{bmatrix} 8 & 0 & 0 & 8 & 8 \\ 0 & 0 & 0 & 8 & 8 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 8 \end{bmatrix} and w1=[80000],w2=[01001],w3=[08000],w4=[−10010],w5=[00100].w_1 = \begin{bmatrix} 8 \\ 0 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad w_2 = \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \\ 1 \end{bmatrix}, \quad w_3 = \begin{bmatrix} 0 \\ 8 \\ 0 \\ 0 \\ 0 \end{bmatrix}, \quad w_4 = \begin{bmatrix} -1 \\ 0 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad w_5 = \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \\ 0 \end{bmatrix}. Then Aw1=8w1,Aw2=8w2+w1,Aw3=0,Aw4=w3,Aw5=0.Aw_1 = 8w_1, \quad Aw_2 = 8w_2 + w_1, \quad Aw_3 = 0, \quad Aw_4 = w_3, \quad Aw_5 = 0. (a) Find the Jordan normal form for AA in the basis {wi}i=15\{w_i\}_{i=1}^5. (b) Find the real stable, unstable, and center subspaces of AA. (c) Find the projections PsP_s, PuP_u, and PcP_c in the standard basis. (d) Find exp⁡At\exp At.

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